Limits and Continuity - Examples and Constructions

Infinite limits and limits at infinity describe the behavior of functions as they approach unbounded values or as their arguments grow without bound. These concepts are essential for understanding asymptotic behavior and classifying singularities.

DefinitionInfinite Limits

We write $\lim_{x \to a} f(x) = \infty$ if for every $M > 0$ , there exists $\delta > 0$ such that $0 < |x - a| < \delta \implies f(x) > M$

Similarly, $\lim_{x \to a} f(x) = -\infty$ means for every $M < 0$ , there exists $\delta > 0$ such that $0 < |x - a| < \delta \implies f(x) < M$

These definitions formalize the idea that $f(x)$ grows arbitrarily large (or decreases without bound) as $x$ approaches $a$ . The vertical line $x = a$ is called a vertical asymptote of $f$ .

ExampleInfinite Limit at a Point

Consider $f(x) = \frac{1}{x^2}$ as $x \to 0$ . For any $M > 0$ , we need $\frac{1}{x^2} > M$ , which is equivalent to $x^2 < \frac{1}{M}$ , or $|x| < \frac{1}{\sqrt{M}}$ . Choosing $\delta = \frac{1}{\sqrt{M}}$ works, so $\lim_{x \to 0} \frac{1}{x^2} = \infty$ This confirms that $x = 0$ is a vertical asymptote.

DefinitionLimits at Infinity

We say $\lim_{x \to \infty} f(x) = L$ if for every $\epsilon > 0$ , there exists $N > 0$ such that $x > N \implies |f(x) - L| < \epsilon$

Similarly, $\lim_{x \to -\infty} f(x) = L$ means for every $\epsilon > 0$ , there exists $N < 0$ such that $x < N \implies |f(x) - L| < \epsilon$

The horizontal line $y = L$ is called a horizontal asymptote of $f$ .

ExampleLimit at Infinity

Consider $\lim_{x \to \infty} \frac{2x^2 + 3x + 1}{x^2 + 5}$ . Dividing numerator and denominator by $x^2$ : $\frac{2x^2 + 3x + 1}{x^2 + 5} = \frac{2 + \frac{3}{x} + \frac{1}{x^2}}{1 + \frac{5}{x^2}}$

As $x \to \infty$ , the terms $\frac{3}{x}$ , $\frac{1}{x^2}$ , and $\frac{5}{x^2}$ all approach 0, so $\lim_{x \to \infty} \frac{2x^2 + 3x + 1}{x^2 + 5} = \frac{2 + 0 + 0}{1 + 0} = 2$

Remark

When evaluating limits of rational functions at infinity, the dominant terms (highest powers) determine the behavior:

If degrees are equal, the limit is the ratio of leading coefficients
If numerator has higher degree, the limit is $\pm\infty$
If denominator has higher degree, the limit is 0

DefinitionInfinite Limits at Infinity

We can combine these concepts: $\lim_{x \to \infty} f(x) = \infty$ means for every $M > 0$ , there exists $N > 0$ such that $x > N$ implies $f(x) > M$ . This describes functions that grow without bound as $x$ increases without bound, such as $f(x) = x^2$ or $f(x) = e^x$ .

Understanding these various types of limiting behavior provides a complete picture of how functions behave both locally (near specific points) and globally (as variables tend to infinity).